The only thing I am able to think up about this problem is that the graph has to have exactly 8 edges.

Last edited by anonimnystefy (2014-04-02 08:53:10)

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: Urgent -Discrete Math problems (Graph)

bob bundy wrote:

Maybe 'similar' doesn't mean what I said.

Have you seen the whole post #4?

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: Urgent -Discrete Math problems (Graph)

Yes, but we can start by noting that the graph conatins exactly 8 edges.

Last edited by anonimnystefy (2014-04-03 05:49:22)

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: Urgent -Discrete Math problems (Graph)

bob bundy wrote:

the incident matrix has exactly two ones and 6 zeros in each column.

Why is that?

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: Urgent -Discrete Math problems (Graph)

It is easy to find P. The process is called diagonalizing a matrix and is a rote procedure. My favorite thing in the world, a no brainer! But it may not exist!

???

You should share your great secrets with the world, then!

Diagonalizing, and its related problem of finding eigenvalues and eigenvectors, is a major subject of research for Numerical mathematics. For 5 dimensions or higher, it can only be done by iterative methods.Check out the Wikipedia page "Eigenvalue Algorithm" for more information.

For 2x2 and 3x3, my favorite method is to solve the determinant equation for the eigenvalues, then exploit the Cayley-Hamilton theorem to find the eigenvectors. If you have enough independent eigenvectors to span the space, then you can use them as the columns of the matrix P (but normalize them first, so that P[sup]-1[/sup] = P[sup]T[/sup], which isn't necessary but is a lot easier).

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

Re: Urgent -Discrete Math problems (Graph)

I apologize for being abrasive. I had not intended to be so. I respect the generosity and intelligence you've amply demonstrated in these forums.

But I am not at all behind the times concerning what a CAS can do. My remarks are entirely acurate. Diagonalization by finite means is not possible beyond 4x4 matrices - at least not without involving trancendental functions, (which even if you used, also require iterative means to calculate). This is not a "nobody has figured out how to do it yet", but rather has been proven to be a fundamental limitation.

That said, there are, and have long been, iterative means of finding eigenvalues and eigenvectors for any size of matrix. But this remains an active area of research as people attempt to find new ways that work faster, either in general, or for specific classes of matrices. Most of the algorithms listed on the Wikipedia page were created within the last 30 years, many within the last 10. And there are others that are not listed.

As for size of matrices, I suspect the record is held by Google's Page Rank algorithm, which finds eignenvalues for a matrix whose dimension is in the trillions now, I think (one for each page on the internet). Fortunately, this matrix is very sparse - almost all of its entries are 0, which makes the job a lot easier.

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

Re: Urgent -Discrete Math problems (Graph)

I never said it was impossible. What is impossible is a general procedure for size > 4. One that works for every matrix. There are always special cases where it is possible to find the exact eigenvalues and eigenvectors (for triangular matrices it is trivial, no matter what the size).

Most of the algorithms listed on the Wikipedia page were created within the last 30 years, many within the last 10.

Not good enough. Many CAS algorithms are secret.

And in what way does this somehow counter the point from which my quote was taken, which was that eigenvalue algorithms are an active field of research?

So if we can please move away from talking past each other, I would rather return to the actual problem here:

Having looked up the definition of Adjacency and Incidence matrices. I note that the adjacency matrix is symmetric. If the incidence matrix is similar to it, it must be symmetric as well. Each row in the incidence matrix is a vertex, and each column is an edge. So each column has exactly two 1s. If the matrix is symmetric, each row has only two 1s as well. This means that each vertex is connected to exactly two edges. As a result, every vertex and every edge must lie in simple loop. If the graph is connected, then it has to be simply one big loop of eight vertices connected by eight edges. There is only one such graph.

Edited to add: I see that only in problem 2 does it assume the graph is connected. This broadens the choices:a loop of 8a loop of 6 and a loop of 2a loop of 5 and a loop of 3two loops of 4a loop of 4 and and two loops of 2two loops of 3 and a loop of 24 loops of 2

Last edited by eigenguy (2014-04-03 12:54:28)

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

Re: Urgent -Discrete Math problems (Graph)

Please do not include me in that statement. It may be true for you, I have honestly answered all your concerns.

What is impossible is a general procedure for size > 4.

A general procedure? Just because a compass and straight edge method does not always work what does that mean? Are those the only methods? The fact is I provided evidence, I can not make you look at it.

And in what way does this somehow counter the point from which my quote was taken, which was that eigenvalue algorithms are an active field of research?

I never said it was not.

In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.